Very Computer

Find a DAG in a str. conn. graph

I am looking for a solution to a problem, that I would describe in abbreviated form as:

Find a maximum covering DAG in a strongly connected directed graph.

In more detail, the problem is as follows:

A strongly connected directed graph is a directed graph, where a path exists betwen any pair of nodes. I want to delete a minimum set of arcs (if not possible, a minimal set) to get a DAG, i.e. the set of deleted arcs should have the property, that none of the arcs may be inserted into the graph without introducing a cycle (minimal) and no other set with less cardinality exists (minimum).

I need an algorithm for the problem and, if possible, a reference to an article or book introducing the algorithm and discussing its computational complexity.

I assume, a polynomial algorithm for this problem exists, because it seems to be of similar difficulty as the problem of finding a minimum spanning tree in an weighted undirected graph with arc weights in the real numbers (by setting all arc weights to -1).

I am no expert in graph theory, so I hope there exists a solution well known in the community of graph algorithm experts.